How does one conclude from this, that neutral axis passes through the centroid?

Question

The book I'm following for Strength of materials states that, in a beam with symmetric cross section (about the plane of bending) and in pure bending, the moment of cross sectional area about the neutral axis is zero, i.e.

$$\int_A y\,dA=0$$

where $y$ is the perpendicular distance of an area $dA$ from the neutral axis.

The book then says that, this could only happen when the neutral axis passes through the centroid of section, and I don't understand why.

I have a feeling this might be very trivial to ask, but please bear with me.

Answer 1

The neutral axis intuitively is an axis that if you support a section on its neutral axis on a straight edge the section will balance itself. Or mathematically if we get the area moment about the neutral axis it should give us zero moments otherwise the section will rotate.

The moment of a differential element with an area dA about this axis is its distance (y)* area (dA).

$$M_{dA}=y dA$$

$$\sigma M_{neutral- axis}=0$$

$$\int ydA=0$$

For example, if we have a square obviously, the neutral axis is at its middle and divides the square into 2 rectangular, one above one below the neutral axis.

$$\sigma M=0 \rightarrow A_{upper - rect}* dx- A_{lower-rect}*dx=0$$

$$A/2 * A/4- A/2 * A/4=0$$

Answer 2

In any cross-section, the neutral axis is located at where the bending stress is zero. In the case of asymmetric cross-section, it falls on the center of the section (where the centroid is located) with the areas above and below the neutral axis are equal, and $y_{top} = y_{bottom}$.

Answer 3

So we know that the moment of the cross sectional area about the neutral axis is zero that is,

$$\int_A y\,dA=0$$

where y is the perpendicular distance from the neutral axis of an area $dA$

Let the origin of the coordinate system lie at the center of Neutral Axis. The $y$ in the above equation is the distance measured along the y axis of this coordinate system.

The centroid C, of this cross sectional area will lie somewhere on the y axis. The y coordinate of the centroid is given by,

$$\bar y= \frac{\int_A y\,dA}{\int_A dA}$$

since $$\int_A y\,dA=0$$

$$\bar y=0$$

This suggests that the centroid is at $y=0$, i.e. on the Neutral axis. Conversely, the neutral axis passes through the centroid of the cross sectional area.